7. Obtain the general equation of the equilibrium of fluids,

dp=p(Xdx + Ydy + Zdz);

and the equation of condition that must be satisfied that the equilibrium may be possible.

8. When any fluid mass is in equilibrium, the points of equal density lie in continuous surfaces, which are also surfaces of equal pressure.

Prove that in the earth's atmosphere, considered independently of the diurnal rotation, the conditions of equilibrium are not satisfied, by reason of the irregular distribution of the heat from the sun, and in consequence that winds continually prevail.

10. A cylindrical vessel of given dimensions containing air, revolves about its axis with a given angular velocity; it is required to find the pressure at the surface of the cylinder, and the law of the variation of the pressure from the axis to the surface.

11. Obtain Laplace's formula for the barometric measurement of heights.

12. If fluid of any kind be moving in such a manner that at the same point in space the velocity is constantly the same in quantity and direction, then if X, Y, Z be the forces impressed at any point whose co-ordinates are x, y, z, and v be the velocity at this point,

13. Ascertain the velocity with which air in a large vessel, maintained in a given state of compression, will issue through a small orifice into the atmosphere.

14. Describe the action of the Hydraulic Ram.

15. When a body acted upon by a force tending to a centre, describes a curve in a resisting medium, the velocity at any point is that acquired by falling down one fourth the chord of curvature by the action of the force at that point.

16. If an indefinitely slender column of air, in which

p = a2D (1 + s),

be any how put in motion, and the motion be small, the velocity v

and condensation s, at a distance x from a fixed point, and at a time t reckoned from a given instant, will be given by the equations,

1.

ST. JOHN'S COLLEGE, DEC. 1820.

EXPLAIN the construction and use of a common barometer and its vernier. Construct also a self-registering thermometer.

2. Shew how the specific gravities of solid and gaseous bodies may be determined.

3. A body when put under the receiver of a given air-pump weighs (a) ounces, and after (n) turns weighs (b) ounces. Required the weight of the body in vacuo; and supposing the specific gravity of the body known, determine the density of the air in the receiver at first.

4. A tretrahedron is filled with water. Having given the length of one of its edges, and the pressure on the base, find the pressure on the sides.

5. A given cone filled with water is supported with its axis inclined to the horizon at a given angle. Find on what section parallel to the base, the pressure is a maximum.

6. Given the quantity of air left in a barometer tube before immersion, find the height at which the mercury is supported after immersion. What would be the height if the particles repelled each

7. A diving bell of given dimensions in the form of a hemispheroid, the sections parallel to the base being ellipses, sinks till the water reaches the middle point of its axis; find its depth below the surface.

8. A given frustum of a paraboloid whose density is (d) sinks to a depth (a) in a fluid, How far will a similar and equal frustum, similarly immersed, whose density varies inversely as the distance from the vertex, and at the smaller end is equal to d, sink in the same fluid ?

9. The groin AMPQ is generated by the motion of a variable parabola apm parallel to itself, the section through the axes of the parabolas being a right-angled isosceles triangle ANM. Given the dimensions of the upper surface PMQ, find the time of emptying through a small given orifice at A, the vessel being full, and the axis AN vertical.

10. A given cylinder rests in water with its axis vertical, and two-thirds of it immersed. Suppose half the part extant to be suddenly taken off, find the time of an oscillation.

11. A prismatic vessel filled to a given altitude, has a small given orifice made in the middle of its side. Compare the forces at first, and after n", with which the spouting fluid strikes a vertical plane. The effects being estimated in a direction perpendicular to the plane, whose distance from the vessel is given.

12. Two equal cylinders A, B, whose density is (2d) and altitude (a) are immersed in different fluids, viz. A in a fluid whose density varies as the depth, and B in a fluid of uniform density (d). When connected by a string passing over a fixed pulley they balance in a given position. Supposing B depressed through a space (c), find by how much it must be lengthened to restore the equilibrium.

ST. JOHN'S COLLEGE, DEC. 1827.

1. THE pressure on any surface immersed in a fluid, is equal to the weight of a column of fluid, whose base equals the surface pressed, and altitude the perpendicular depth of its centre of gravity.

2. A body is floating between two known fluids, and the part immersed in the lower is observed to be the same, as if it were floating on the surface of a fluid, formed by the mixture of equal quantities of the two fluids; required the specific gravity of the

solid.

3. A sphere is full of fluid; draw that horizontal section which shall sustain the greatest pressure, and compare that pressure with the pressures on the two surfaces into which the sphere is thus divided.

4. Explain the cause, and action of reciprocating springs,

5. A given cylinder is excavated beneath into the form of an hemispheroid, having the same base and altitude. How high may this be filled with fluid, whose specific gravity equals n times that of the cylinder, before it begins to raise it from the horizontal plane?

6. A cone with its density varies as the

vertex upwards is filled with fluid, whose nth power of the perpendicular depth, required to compare the pressure on the conical surface with the weight of the fluid.

7. Find the resistance on an oblate spheroid moving in a fluid in the direction of its axis.

8. To find the effect on the graduation of a barometer tube, when the area of the basin is taken into account.

9. A vessel of fluid in the form of a paraboloid is placed with its vertex downwards, find where a small given orifice must be made, that the fluid may continue to issue for the greatest length of time, and compare that time with the time of emptying the whole vessel through the same orifice in the lowest point.

10. The barrel of an air-pump discharges at every stroke into the receiver of a condenser. Required the density in the condenser after n strokes, both vessels being filled with common air at first. Also determine the limit to the increase of density in the condenser. 11. There is a flood-gate communicating with a reservoir of water, in the form of a rectangle, moveable above its upper edge, which is horizontal, and at a given perpendicular depth below the surface of the fluid. Given the weight and dimensions of the flood-gate; required at what angle it must be inclined to the horizon, that it may confine the water by its weight alone.

12. A cylindrical vessel of air revolves above its axis, so that the velocity of its circumference equals that acquired down n times the height of an homogeneous atmosphere. Required the density of the air in the cylinder at different distances from the centre, neglecting the effect of gravity on the air. And shew that when n is very small, the distance of the point where the density remains the same,

1

as when the cylinder was at rest, equals radius ultimately.

√2

F

ST. JOHN'S COLLEGE, DEC. 1828.

The

1. DEFINE W, M, and S, and prove that W = MS. specific gravities of pure gold and copper are 19.3 and 8 62. Required the specific gravity of standard gold, which is an alloy of eleven parts pure gold and one part copper.

2. A sphere floats between two fluids of known specific gravity; given the ratio of the portions of surface immersed in each fluid, find the specific gravity of the sphere.

3. A tube of given length is inserted in the side of a vessel of fluid at a given depth from the surface. What must be the inclination of the tube that the latus-rectum of the parabola described by the issuing fluid may be a maximum?

4. If the particles of air repel each other with a force varying inversely as the nth power of the distance of their centres; r = radius of the Earth, h and H the altitudes of a homogeneous atmosphere at the Earth's surface, and the distance x above it, shew that

5. Construct and explain the action of the air-pump.

6. Explain how a rudder acts upon a boat. Give its greatest effect in altering the course; how much is the boat then resisted by it?

7. A body is projected against a stream whose resistance α v", with a velocity equal to that of the stream. Required the space described before the body begins to return; and shew that, when 2, this space is the same for streams flowing with different velocities.

n=

8. Determine the length of a gun, that the velocity of the discharge may be a maximum, supposing the friction of the barrel to be

constant.

⚫ 9. The vertex of a conical body whose vertical angle is inconsiderable is fixed in a fluid at rest at a depth below the surface less than the length of the cone, and in a known ratio to it. Given the relative specific gravities of the solid and fluid, required the position of equilibrium?